Generalized jordan curve theorem

[StatusX]

Is anyone here familiar with the proof (using homology) of the generalized Jordan curve theorem, that a subspace of S^n homeomorphic to S^(n-1) divides it into two components? It can be found on page 169 of Hatcher's algebraic topology book, which can be downloaded from http://www.math.cornell.edu/~hatcher/AT/ATpage.html" page.

I'm wondering if it's possible to use the same kind of proof to show that a general (n-1)-manifold divides S^n into two components. It was shown that the complement of an n-1-sphere in S^n actually has the homology of S^0, which is much stronger, and won't hold for general n-1-manifolds. But all I need is that the 0th reduced homology group is Z, which does seem to be true. The problem is that the corresponding terms of the Mayer-vietoris sequence used in Hatcher's proof aren't zero where they need to be. Does anyone have any ideas?

 

[mathwonk]

are you using alexander duality? see p.254.

 

[StatusX]

I haven't gotten to that yet (Im just starting cohomology), though I see how it would follow from there. Is there an easier method, or a way to extract that bit of information using the idea of alexander duality without developing all the machinery?